Paul Raeburn is a co-author of "The Game Theorist's Guide to Parenting" and the author of "Do Fathers Matter?"

This month, the Supreme Court struck down a law that treated unwed mothers and fathers differently when granting citizenship to their children born outside the United States — the requirements for fathers were stiffer. Justice Ruth Bader Ginsburg, authoring a majority opinion joined by five other justices, wrote that the law was based on gender stereotypes that violated the notion of equal protection. The law implied “that unwed fathers care little about, indeed are strangers to, their children.” One might think that idea had been discarded long ago. Unfortunately, many such myths about fathers persist.

Myth No. 1

Men can become fathers at any age.

We’re all familiar with the risks of older motherhood. But older fathers often get a pass. “There are some things [men] never had to worry about. Like the ticking of the biological clock,” The Washington Post’s Richard Cohen wrote in the 1978 column that coined that term. Sometimes, older fathers are even counted as a plus. “Fathers who are older typically have more leisure time, more financial resources, and seem to be more thoughtful and proactive” than their younger counterparts, according to the National Center for Fathering.

It’s not that simple. Men’s fertility declines as they age — although more slowly and somewhat later than women’s. Decreasing testosterone levels and a drop in the quantity and quality of sperm can make it harder for men in their 40s and older to conceive. And the children of older fathers face higher risks for a variety of genetic conditions. It’s well known and understood that older women have an increased likelihood of having a child with Down syndrome. Here’s the surprise: A 40-year-old man’s risk of having a child with Down syndrome is the same as a 40-year-old woman’s. Furthermore, the child of a 40-year-old father has a 2 percent chance of having schizophrenia — double that of children whose fathers are under 30. And children of fathers over 50 have a nine-fold increase in the risk of autism compared with children of dads younger than 30.

Myth No. 2

Fathers don’t feel pressure to have it all.

The “having it all” discussion almost always centers on mothers, as in Anne-Marie Slaughter’s influential Atlantic piece, “Why Women Still Can’t Have It All.” Slaughter wrote that men don’t seem to get the same grief for choosing work over family. Meanwhile, research shows that there is a real motherhood penalty, in terms of pay and hiring potential, that doesn’t translate to fathers. Indeed, men with children tend to earn more than childless men, and they’re seen as more desirable hires.

Yet Ellen Galinsky of the Families and Work Institute says fathers are being hurt by a “male mystique” that is as difficult as the feminine mystique was for women.

Galinsky’s National Study of the Changing Workforce found that while work and family conflicts have remained fairly steady for mothers over the past three decades, the share of fathers reporting work-family conflicts rose from 35 percent in 1977 to 60 percent in 2008. Men with children say they feel continued pressure to be the primary providers for their families (in opinion polls, about two-thirds of Americans say a married man should be able to support his family), and at the same time they want to meet modern fathering ideals (in polls, they are just as likely as mothers to say that parenting is “extremely important” to their identity). Even when flexible schedules and other family-friendly work arrangements are available to men, there’s often a stigma associated with taking advantage of them. According to a report by the Center for American Progress, “Men who request workplace flexibility for family reasons receive lower wages, poorer performance evaluations, and fewer promotions than their counterparts who maintain regular work schedules.” In other words, they get penalized much as mothers do.

Myth No. 3

Fathers are dispensable.

Jane Mattes, the founder of Single Mothers by Choice, is among those who think we can do without dads. “We probably would all agree that a child needs at least one consistent, stable and mature parent who can be loving and is able to set limits,” she wrote in an opinion piece a few years ago. “This could be a single mother or a single father.”

This is a obviously a claim with immense political implications. So it’s important to parse the issues clearly. Yes, healthy and happy children are being raised by single mothers, and that’s been true for a long time. But fathers contribute things to children’s well-being that cannot be easily replaced.

Fathers play important roles in children’s development, from gestation onward. For example, infants whose fathers were involved during pregnancy are less likely to be born prematurely or with a low birthweight — both substantial risks to health. Conversely, the death rate of infants whose fathers were not around during pregnancy is nearly four times that of infants whose fathers were present, controlling for other factors such as poverty.

Children whose fathers play with them, read to them, take them on outings and help care for them have been shown to have higher IQs, fewer behavioral problems in the early school years, and less likelihood of delinquency or criminal behavior as adolescents. Children with involved fathers are less likely to smoke and less likely to suffer from depression or other psychiatric ailments years later.

Fathers also make unique contributions to language development in their children. Indeed, in this regard, fathers matter more than mothers. When dads used more words with preschool children during play, the children had more advanced language skills a year afterward. That is likely to lead to increased success in school later on.

Myth No. 4

Fathers don’t connect with children the way mothers do.

Women are often imagined to possess “maternal instincts,” whereas men are told they must work to become good parents and connect with their children. The Internet is loaded with instructions for fathers, who presumably need a manual. “You’re a dad now!” says one site. “It’s exciting, though parts of it may be new to you.”

The reality is quite different, as research has shown. Fathers, like mothers, experience a variety of hormonal changes during pregnancy. One such change for dads-to-be is a drop in testosterone — possibly reflecting a shift from competitive mate-seeking to a more nurturing posture.

In one study, men who were exposed to their newborn children for only 60 minutes were able to recognize their infants by simply touching their hands. Fathers also quickly learn to distinguish their baby’s cry from that of other children.

Everyone’s heard the cliche: “Just wait until your father comes home.” Researchers have found that fathers tend to be more direct and firmer than mothers in their disciplinary approaches, reinforcing their reputation as enforcers. In their book “Partnership Parenting,” Kyle Pruett and Marsha Kline Pruett outline how fathers are often more willing than mothers to confront their children and less willing to rationalize or negotiate punishments with them.

But, as the Pruetts note, mothers are still responsible for the majority of child care and thus are more often the ones setting limits and enforcing discipline. And sometimes, when children defy those limits, it may be that they “have gotten too familiar with, or deaf to, her resolution tactics over time.”

Rather than seeing their fathers as disciplinarians, children from a very young age often associate fathers with play. Studies have found that whereas much of mothers’ interactions with their kids involve routine tasks — getting ready in the morning, driving carpools, preparing dinner — dads spend much of their child-care hours doing things like rough-housing or playing catch with their kids. Child development specialist T. Berry Brazelton describes how babies as young as 4 weeks old react to their dads: “With the father, the baby’s facial features all go up, the extremities tense and waiting, as if the new baby had already learned that his father plays with him.”